The ${\mathrm{E}}_2^{h{C}_6}$-homology of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$

Let ${\mathrm{E}}_2$ be the Morava E-theory of height 2 at the prime 2. In this paper, we compute the homotopy groups of ${\mathrm{E}}_2^{h{C}_6}\wedge \mathbb{R}{P}^2$ and ${\mathrm{E}}_2^{h{C}_6}\wedge \mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$ using the homotopy fixed point spectral sequences.